Topology, Geometry, and Group Theory

9803374戴维斯大约50年前，亚历山大、布斯曼和沃尔德提出了定义比黎曼流形更一般的度量空间的“曲率”的上界和下界的想法。亚历克桑德罗夫主要研究非负曲率的情形及其在欧氏空间中凸多面体曲面上的应用。大约十年前，格罗莫夫指出，由于拓扑学和群论的原因，非正曲线的情况比正曲线的情况有趣得多，这主要是因为非正曲线空间的泛覆盖是可压缩的，这重新引起了人们对这个问题的兴趣。格罗莫夫还指出，这样的空间有很多多面体的例子。尤其是乳房建筑就是这样的例子。在过去的几年里，查尼和戴维斯教授在研究非正弯曲多面体及其在群论中的应用方面进行了合作。他们主要感兴趣的集团是Coxeter集团和Artin集团。这两种类型的组都与反射生成的组相关联。查尼和戴维斯也在这些领域分别完成了研究。他们将继续在这些领域以及密切相关的主题上进行研究，如Tits建筑的性质及其自同构群。反射产生的基团存在于数学和自然界中的许多地方(例如，在结晶学中)。欧几里得平面的正则平铺的对称群就是这样的群。埃舍尔的画展示了非欧几里得平面几何中的规则平铺的例子。这些也与反射组相关联。查尼和戴维斯教授对在更抽象的背景下的这样的例子感兴趣。结果表明，抽象反射群(Coxeter群)总是可以被实现为非正弯曲空间的对称群，就像欧几里得平面和非欧几里得平面的情况一样。***

9803374 Davis About 50 years ago, Aleksandrov, Busemann and Wald introduced the idea of defining upper and lower bounds for the ``curvature'' of more general metric spaces than Riemannian manifolds. Aleksandrov was interested primarily in the case of nonnegative curvature and its applications to convex polyhedral surfaces in Euclidean 3-space. Interest in this subject was renewed about ten years ago when Gromov pointed out that for topological and group theoretic reasons the nonpositively curved case was much more interesting than the positively curved case, the main reason being that the universal cover of a nonpositively curved space is contractible. Gromov also pointed out that there were many polyhedral examples of such spaces. In particular, Tits buildings are such examples. For the past several years, Professors Charney and Davis have collaborated in investigating nonpositively curved polyhedra and applications to group theory. The groups in which they are primarily interested are Coxeter groups and Artin groups. Both types of groups are associated to groups generated by reflections. Both Charney and Davis have also accomplished separate research in these areas. They will continue their research on these areas as well as on closely related topics such as the properties of Tits buildings and their automorphism groups. Groups generated by reflections occur in many places in mathematics and in nature (for example, in crystallogrophy). The symmetry groups of regular tilings of the Euclidean plane are such groups. Escher's drawings show examples of regular tilings in non-Euclidean plane geometry. These are also associated with reflection groups. Professors Charney and Davis are interested in such examples in a more abstract setting. It turns out that abstract reflection groups (Coxeter groups) can always be realized as the symmetry groups of a space that is nonpositively curved, much as in the case of the Euclidean and non-Euclidean planes. ***